Michel Planat

Multiple scale analysis of the nonlinear surface acoustic wave propagation in anisotropic crystals

A fundamental theory of nonlinear Rayleigh wave propagation is given here. Previously, the harmonic boundary value problem was solved by means of a direct iterative procedure. The nonlinear acoustic field could only be studied near the excitation transducer. The method could not explain the nonmonotonous behavior of curves giving the amplitude of various harmonics as a function of the propagation distance. In this paper, amplitude and phase relations of harmonics are calculated also in the far field. Anisotropy effect is demonstrated in the first numerical results obtained for various quartz crystal cuts.

Efficiency of surface interdigital transducers on anisotropic media

Excitation and detection of surface acoustic waves by means of interdigital transducers are studied here in the case of general anisotropic media with low piezoelectric coupling. Relations between mechanical wave amplitude and applied and detected voltage are established. The electromechanical coupling coefficient is calculated from the equivalent electrical circuit of the transducer for different quartz‐crystal orientations. Experimental results are given for different quartz‐crystal cuts, in agreement with the theoretical values.

Generation of 1/f noise in locked systems working in nonlinear mode

Properties of the phase fluctuations of a nonlinear oscillator were investigated. The multiscale aspect of synchronization is experimentally confirmed with the observation of lockings at any rational ratio. This global property of synchronization is associated with a local one by the study of the phase locked loop (PLL) under nonlinear conditions. We show that the discontinuity between the locking and the free-run state is characterized by the presence of slow temporal variations that induce a low frequency spectrum. An experimental verification of the increase of fluctuations confirms the classical results, which associate 1/f noise to nonlinearity, autosimilarity and regularity, represented here by the synchronization.

FREQUENCY STABILITY ENHANCEMENT FROM CARRIER-ENVELOPE RESONANCE IN A SURFACE ACOUSTIC WAVE DELAY LINE OSCILLATOR

We have built a new injection‐locking set‐up which allows the study of low frequency noise in an oscillator. The interaction between a driving 100 MHz signal and a surface acoustic wave delay line oscillator undergoing a nonlinear modulational instability is investigated. Driving amplitude is used to control the synchronization range and therefore the frequency stability of a newly discovered carrier‐envelope resonance process. In the soliton‐locked regime, fundamental and subharmonic steps attached to a Lorentzian shape frequency‐amplitude characteristic are found. They are explained thanks to a driven nonlinear Schrodinger model and the joined synchronization map.

Close to resonance interaction of radiofrequency waves in a Schottky diode mixer: 1/f noise and number theory

1/f noise arising from the mixing and phase locking of two input radio frequency oscillators in a doubly balanced phase bridge with Schottky diodes is investigated. It is interpreted from number theoretical concepts using continuous fraction expansions of the frequency ratio of signals at the input of the mixer. Ingredients of the theory are a diophantine approximation as a way to compute beat signals, a Farey tree to control the strength of intermodulation products, and the ordering of the leaves in the tree with respect to the uniform distribution as a way to express the phase locking between the oscillators and to compute the associated 1/f spectrum. Such a mechanism relates to Franel-Landau conjecture which is one alternative to Riemann’s hypothesis. Almost resonant states in the mixing of oscillators are revealed experimentally as frequency jumps of the beat signal (a random telegraph type signal) and are seen as the phase locked states of the theory.

Effect of piezoelectricity on the excitation and radiation of acoustic waves in rotated Y-cut quartz

Recently we reported a far field theory of transverse acoustic waves radiated from an interdigital transducer deposited on a singly rotated quartz crystal. This was done by an approximation called the local decoupling approach, allowing us to consider electrical effects as forcing fields for the mechanical motion. The present analysis demonstrates that this approximation is not valid in predicting the complete spectrum of acoustic waves. It is shown that near the cutoff frequency the bulk wave may be converted into a surface wave of Bleustein–Gulyaev type in the free surface case, and that a surface metallization reinforces the strength of this conversion. Velocity of surface waves are determined by means of the effective permittivity function and correct radiation diagrams are plotted in the far field.

Reflection of surface acoustic waves on anisotropic media

Reflection of Rayleigh surface waves by shallow grooves or metallic strips on general anisotropic crystals is analyzed. In this paper, the groove or strip is considered as a small discontinuity which acts as a surface excitation source for the scattering of the incident wave. The problem is solved by using Fourier transform technique and integration in the complex plane. Reflection coefficients of the various geometries on quartz and lithium niobate are calculated in good agreement with experiment.

The arithmetic of 1/f noise in a phase locked loop

Frequency countings close to a phase locked zone in an electronic receiver show a 1/f power spectral density. The noise scaling versus the frequency deviation and the open loop gain is found from Adlers model of the phase locked loop. This fully agrees with experiments performed at 5 MHz on a receiver with a Schottky diode mixer and a low pass filter. The 1/f amplitude and frequency noise due to the whole set of (sub)harmonics is explained from a nonlinear mapping, with a coupling coefficient related to the structure of prime numbers. The number theoretical approach of 1/f noise and the link to the generalized Riemann hypothesis is explained.

TEMPERATURE DEPENDENCE OF 1/f NOISE IN QUARTZ RESONATORS IN RELATION WITH ACOUSTIC ATTENUATION

The 1/f noise behaviour of quartz crystal resonators between liquid helium and room temperature is investigated in relation with acoustic attenuation. Large attenuation is observed at 20 K and 50 K The 50 K is due to ionic Na + impurity relaxation and the 20 K peak to the interaction of the sound wave with thermal phonons. A strong correlation between 1/f noise level and the acoustic attenuation is found. These results support for quartz crystal the hypothesis of the origin of 1/f noise in phonons fluctuations, and principally in the fluctuations of their relaxation time. The relation between 1/f noise and Q-factor follows a 1/Q 4 law as it has already been observed at room temperature with resonators of various frequencies. This correlation is not observed on the Na + impurity relaxation peak. At very low temperature (below 10 K), the attenuation increases again and 1/f noise level follows approximately the same Q-factor dependence.

Binary coding and correlations in the noise frequencies of an oscillator

We report on a multifractal type approach to the study of low frequency noise in oscillators. The time sequence is interrogated by means of a multiscale local stability exponent which acts as a sliding filtering window. Its discontinuities are reflected into a binary coding of which the correlations are analyzed. The mapping of binary data into a devil’s staircase is used to emphasize the correlated origin of frequency fluctuations.